On the Classical Prime Radical Formula and Classical Prime of Semimodules
Keywords:prime subsemimodule, classical prime subsemimodule, prime radical, classical prime radical, prime ideal
Let $R$ be a commutative semiring and $M$ an Â $R$ semimodule. A proper subsemimodule Â $N$ of Â $M$ is called a classical prime subsemimodule, if for any Â $a,b\in R$ and Â $m\in M, abm\in N$ implies that Â $am\in N$ or $bm\in N$. Â We will introduce and study the notion of prime bases for classical prime subsemimodules and utilize them to derive some formulas on the classical prime radical of subsemimodules of a semimodule. In particular, we study some basic properties of prime radical and classical prime radical of subsemimodule in Â $M$. Moreover, we investigate relationships between classical prime radical and prime radical of subsemimodule in $M$.
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